Subtraction
Subtraction is really just a specific example of the distributive property that we saw in the previons section. However, it seems to cause students a lot of trouble so we wanted to devote an entire section to it. I'll explain the procedure by working through an example.
| Simplify: | $(3x^2 + 2x - 4) - (-3x^2 + x - 2)$ |
| Remember that the minus sign is like having a negative one in front of the expression. We can write that explicitly this way. | $(3x^2 + 2x - 4) - 1(-3x^2 + x - 2)$ |
| Now I'll apply the Distributive Property that we saw in the last section to distribute the -1 into the parentheses. | $(3x^2 + 2x - 4) + (-1)(-3x^2) + (-1)x + (-1)(-2)$ |
| From here we simplify the problem just like the ones in the last section. First I'll do the arithmetic with the coefficients. | $(3x^2 + 2x - 4) + 3x^2 + -x + 2$ |
| Now I'll group the like terms together. | $3x^2 + 3x^2 + 2x + -x + 2 - 4$ |
| And finally I'll combine them to get the final answer. | $6x^2 + x - 2$ |
That's the formal way of doing the simplification but it always seems like a lot of work to me. With all those - and + signs floating around, there's just too many opportunities for something to go wrong. I like to think of it this way: To distribute the negative sign, just reverse the signs of every term inside the parentheses. Here's a few more examples.
Quick Tip - Checking Your Answer
There's no way to completely check your answer with this kind of problem but there is a method you can use that will let you increase your confidence with your answer. First pick a value for x. Then substitute that value into both the original expression and your final, simplified version. If you don't get the same number from both versions, you know you made a mistake. If you get the same number, it doesn't guarantee that your answer is right - you may have just been really unlucky and picked a number that just happens to be the same in both versions but this should make you a little more comfrotable with your answer. (You can take it even further by checking more values. The more matches you get the more likely yoru answer is the right one.)
Here's how it would work with Example 1. Say I pick x = 3. If I substitute this into the original expression, I get:
(2(3)2 + 2) - (3(32) - 3 + 2)
(18 + 2) - (27 - 3 + 2)
20 - 26
-6
Now if I make the same substution into my simplified version, I get:
-(32) + 3 = -9 + 3 = -6
Since I got the same number both times, I'm pretty confident my solution is correct.
Example 1
Simplify (2x2 + 2) - (3x2 - x + 2).
First, I'm going to distribute the negative sign into the parentheses by reversing the sign of every term inside them.
(2x2 + 2) - 3x2 + x - 2
Notice how the first term became negative. In the original problem, it's a +3x2 so the negative sign made it -3x2. Now I'll combine the like terms:
2x2 + 2 - 3x2 + x - 2 + 2
And finally, I'll combine the like terms together.
(2 - 3)x2 + x - 2 + 2 = -x2 + x
Example 2
Simplify (x3 + x + 1) - 3(-4x2 + 2).
This problem is similar to the last one in that they both involve subtraction but this one has that additional three in front of the second parentheses. There are a number of ways you can handle this sort of situation. I'm going to do it in two steps: First, I'll distribute the three then I'll deal with the negative sign.
(x3 + x + 1) - 3(-4x2 + 2)
(x3 + x + 1) - (3·(-4x2) + 3·2)
(x3 + x + 1) - (-12x2 + 6)
Notice how the only thing I moved was the 3. The minus sign stayed on the outside. Now I'm going to move the negative sign inside the parentheses by changing the sign of everything inside them.
(x3 + x + 1) + 12x2 - 6
Now I'll group the like terms and combine them to get the final answer.
x3 + 12x2 + x + 1 - 6
x3 + 12x2 + x - 5
